for every \(x\in G\)  and \(s\in H\) .
The covariant functions appear in abstract harmonic analysis in the construction of induced representations, see [1]}, [2]}. We here employ some of the classical tools in this direction. Suppose that \(\lambda _H\)  is a left Haar measure on \(H\) .
For each character \(\xi \in \chi (H)\)  and a function
\(f\in \mathcal {C}_c(G)\) , define the function \(T_\xi (f):G\rightarrow \mathbb {C}\)  via
\(T_\xi (f)(x):=\int _Hf(xs)\overline{\xi (s)}\mathrm {d}\lambda _H(s),\) 
